A von Neumann–Morgenstern Representation
Result without Weak Continuity Assumption
Freddy Delbaen∗◦
Samuel Drapeau‡
Michael Kupper §
April 17, 2011
In the paradigm of VON N EUMANN AND M ORGENSTERN, a representation of affine preferences in terms of an expected utility can be obtained under the assumption of weak continuity. Since the weak topology is coarse, this requirement is a priori far from being negligible.
In this work, we replace the assumption of weak continuity by monotonicity. More precisely,
on the space of lotteries on an interval of the real line, it is shown that any affine preference
order which is monotone with respect to the first stochastic order admits a representation in
terms of an expected utility for some nondecreasing utility function. As a consequence, any
affine preference order on the subset of lotteries with compact support, which is monotone
with respect to the second stochastic order, can be represented in terms of an expected utility for some nondecreasing concave utility function. We also provide such representations
for affine preference orders on the subset of those lotteries which fulfill some integrability
conditions. The subtleties of the weak topology are illustrated by some examples.
Key Words: Affine Preference Order; von Neumann-Morgenstern Representation; Automatic Continuity; First Stochastic Order; Weak Continuity
Introduction
The problem of assessing uncertain monetary ventures by way of an expected utility dates back to the
early stages of probability theory with the works of B ERNOULLI and C RAMER on the St. Petersbourg
paradox. However, in an approach consisting primarily in the study of preferences on lotteries, VON
N EUMANN AND M ORGENSTERN [15] provided in their seminal work the first formal systematic analysis
of expected utility. These preferences are specified in terms of a preference order1 < which is required to
∗ The
authors thank two anonymous referees for valuable comments and suggestions. The authors further thank Patrick Cheridito,
Hans Föllmer and Fabio Maccheroni for helpful comments and fruitful discussions.
◦ [email protected]; ETH Zurich, CH-8092 Zurich, Switzerland.
‡ [email protected]; Humboldt University Berlin, Unter den Linden 6, D-10099 Berlin, Germany. Financial
support from the DFG IRTG 1339 and from MATHEON project E.11 is gratefully acknowledged.
§ [email protected]; Humboldt University Berlin, Unter den Linden 6, D-10099 Berlin, Germany. Financial support from MATHEON project E.11 is gratefully acknowledged.
1 A preference order is a transitive and complete binary relation <. As usual, the notation and ∼ describe respectively the
antisymmetric and equivalence relations.
1
satisfy the independence and Archimedean properties2 . They show that such preference orders < admit
an affine numerical representation U which is unique up to strictly monotone affine transformations.
Preference orders admitting an affine numerical representation will be here referred to as affine preference
orders. Under the additional condition that < is weakly continuous3 , the affine numerical representation
U can be expressed in terms of an expected utility
Z
U (µ) = u dµ,
(0.1)
for some continuous bounded utility function u, generically called von Neumann–Morgenstern representation.
The independence and Archimedean properties were disputed in several subsequent studies and extensions4 , but we here want to focus on this last requirement of weak continuity. In addition to being a strong
mathematical requirement since the weak topology is coarse, it is actually truly questionable from a normative viewpoint; in contrast to the Archimedean property which is empirically as problematic to test but
still has a certain normative appeal5 , the weak continuity condition is primarily a technical one. Starting
with this observation as bottom line, the goal is to give another normatively satisfying condition which
ensures the weak continuity of the preference order, and in turn, provides a representation of the form
(0.1). The condition we consider is a monotonicity with respect to the first or second stochastic order, the
normative and theoretical interest of which in the context of preferences has been studied by H ADAR AND
RUSSELL [9]. The works of NAMIOKA [14] and B ORWEIN [3] include results when the monotonicity
of convex functions on metrizable vector spaces induce automatically their continuity. These results were
already mentioned and partly exploited in D RAPEAU AND K UPPER [5] in the context of general risk
orders. However, these classical results on automatic continuity cannot be directly applied here. Indeed,
the space of lotteries is a convex set and not a vector space. Moreover, the weak topology on the vector
space of signed measures spanned by the lotteries is not metrizable.
To overcome these problems, we first show that any affine preference order which is monotone with
respect to the first stochastic order is continuous for the variational norm. This does not imply the weak
continuity of the preference order, mainly because lotteries cannot be approximated in the variational
norm topology by simple lotteries6 . There are actually examples of affine preference orders monotone
with respect to the first stochastic order which are not weakly continuous. However, our second main
result shows that these affine preference orders monotone with respect to the first stochastic order admit
a von Neumann–Morgenstern representation in terms of a nondecreasing utility function. In this context,
in spite of such a representation, the existence of a certainty equivalent is not necessarily ensured. Such
a representation result does not hold for affine preference orders on arbitrary subsets of lotteries. However, we obtain similar representation results for the subspace of lotteries with compact support, showing
in particular that monotonicity with respect to the second stochastic order is enough to ensure a von
Neumann–Morgenstern representation. We also extend these results to some other canonical subspaces.
The structure of this paper is as follows. In the first section we present the setting and state the main
representation results for affine preference orders on the set of lotteries. In the second section, we extend
2A
preference order < satisfies the independence property if for any two lotteries µ, ν with µ ν holds αµ + (1 − α) η αν + (1 − α) η for any lottery η and α ∈ ]0, 1[. It satisfies the Archimedean property if for any three lotteries µ, ν, η with
µ η ν there exist α, β ∈ ]0, 1[ such that αµ + (1 − α) ν η βµ + (1 − β) ν.
3 A preference order is weakly continuous if the sets L (µ) := {ν | µ < ν} and U (µ) := {ν | µ 4 ν} are weakly closed for any
lottery µ.
4 For an overview of which, we refer to F ISHBURN [6].
5 In the sense that it is merely a one dimensional regularity assumption of the preferences along a segment between two alternatives.
6 Convex combinations of Dirac measures.
2
these representation results for the particular subspaces of those lotteries with compact support as well
as those which satisfy additional integrability conditions. In an appendix, we first give some standard
definitions and discuss a Banach lattice structure for the first stochastic order. For the sake of readability,
all the proofs are postponed to the Appendix.
1. Automatic Continuity and Main Representation Results
In the sequel, the affine preference order7 < is defined on the set of lotteries M1 := M1 (S), that is, the
set of probability distributions on the real line with support on a given interval S ⊂ R. As mentioned in
the introduction, we do not require the weak σ (M1 , Cb )-continuity8 of the preference order to obtain a
von Neumann–Morgenstern representation. We instead require the monotonicity with respect to the first
stochastic order. The first stochastic order, denoted by Q1 , is defined by
Z
Z
µ Q1 ν ⇐⇒
u dµ ≥ u dν for any nondecreasing u ∈ Cb .
A preference order < is monotone with respect to the first stochastic order if µ Q1 ν implies µ < ν.
We first present an automatic continuity result for the variational norm k·k.
Proposition 1.1. Let < be an affine preference order on M1 which is monotone with respect to the first
stochastic order. Then, the preference order < is k·k-continuous9 . In particular, any affine numerical
representation U of < is k·k-continuous, and there exists M > 0 such that for any µ, ν ∈ M1 ,
|U (µ) − U (ν)| ≤ M kµ − νk .
(1.1)
The proof in Appendix B.1 is based on ideas by NAMIOKA [14] and B ORWEIN [3], whereby M1 is a
convex set. On the other hand, an automatic σ (M1 , Cb )-continuity proof along this line is not possible.
Indeed, even though M1 is metrizable for the weak σ (M1 , Cb )-topology, the vector space of signed
measures denoted by ca, is no longer metrizable. For S = R, the preference order corresponding to the
affine numerical representation
Z
U (µ) = 1[0,1] (x) + 21]1,+∞] (x) µ (dx) ,
(1.2)
is monotone with respect to the first stochastic order but is neither σ (M1 , Cb )-upper semicontinuous nor
σ (M1 , Cb )-lower semicontinuous. Notice that this example is however continuous for the finer topology
σ (M1 , Bb ) induced by the bounded functions of finite variation10 Bb := Bb (S). In fact, the previous
automatic continuity result holds for this topology as stated in our second main result.
Theorem 1.2. Let < be an affine preference order on M1 which is monotone with respect to the first
stochastic order. Then, the preference order < is σ(M1 , Bb )-continuous and it admits a von Neumann–
Morgenstern representation
Z
U (µ) =
u dµ,
µ ∈ M1 ,
(1.3)
for some bounded nondecreasing utility function u : S → R.
7 Recall
that a preference order is affine exactly when it satisfies the independence and Archimedean properties.
Cb denotes the vector space of continuous bounded functions u : S → R. For the definition of the weak continuity, we
refer to the Appendix A.
9 Both L (µ) := {η | µ < η} and U (ν) = {η | η < ν} are k·k-closed for any µ ∈ M .
1
10 A function is of finite variation if it is the difference of two nondecreasing functions.
8 Here,
3
The proof of this theorem is treated in the Appendix C.1.
Remark 1.3. In the 1930s, KOLMOGOROV [12], NAGUMO [13], DE F INETTI [4] and H ARDY, L ITTLE WOOD , AND P ÓLYA [10, Proposition 215, Page 158] studied the notion of means and obtained under the
assumption of strict monotonicity with respect to the first stochastic order and the existence of a certainty
equivalent11 a representation of the form (1.3) in terms of a continuous increasing utility function u. Theorem 1.2 extends their representation results in the following sense. Firstly, S is not necessarily compact
and we are not restricted to the set of simple lotteries. Secondly, we do not assume strict monotonicity
and finally, the existence of certainty equivalent is not required. Note that a von Neumann–Morgenstern
representation (1.3) does not necessarily imply the existence of a certainty equivalent12 . For instance, the
image of U in (1.2) is equal to the interval [0, 2] whereas the image of its restriction to the set of Dirac
measures is equal to the triple {0, 1, 2}.
Remark 1.4. Monotonicity with respect to the first stochastic order is different from the sure thing principle13 under which F ISHBURN [7] provided a von Neumann–Morgenstern representation. For instance,
the preference order in Example 2.1 is monotone with respect to the first stochastic order whereas the
sure thing principle does not hold.
Remark 1.5. F ÖLLMER AND S CHIED [8, E XAMPLE 2.27] give an example of a non monotone affine
preference order < which does not have a von Neumann–Morgenstern representation.
Under a regularity condition on a subset of simple lotteries, we obtain the σ (M1 , Cb )-upper semicontinuity of the preference order.
Corollary 1.6. Let < be an affine preference order on M1 fulfilling the same assumptions as in Theorem
1.2. Suppose furthermore that for any t, t0 ∈ S and λ ∈ [0, 1] the set
n
o
s ∈ S δs < λδt + (1 − λ) δt0
(1.4)
is closed in S. Then, the preference order < is σ(M1 , Cb )-upper semicontinuous and it admits a von
Neumann–Morgenstern representation
Z
U (µ) = u dµ, µ ∈ M1 ,
(1.5)
for some bounded nondecreasing right-continuous utility function u : S → R.
The proof of this corollary is treated in the Appendix C.2.
Remark 1.7. The assumption (1.4) only ensures the σ (M1 , Cb )-upper semicontinuity, in contrast to the
standard automatic continuity results which guarantee the continuity. Indeed, for S = R, the preference
order corresponding to the numerical representation
Z
U (µ) = 1[0,+∞[ (x) µ (dx) ,
is σ (M1 , Cb )-upper semicontinuous but not σ (M1 , Cb )-continuous. This example again shows that the
von Neumann–Morgenstern representation (1.5) does not necessarily have a certainty equivalent.
11 An
element µ ∈ M1 admits a certainty equivalent if there exists c ∈ S such that δc ∼ µ.
that the existence of a certainty equivalent is a strong requirement which even fails if u is strictly increasing with jumps.
13 If µ is a lottery, A is a Borel set with µ (A) = 1 and s ∈ R, then δ < µ if δ < δ for all t ∈ A and µ < δ if δ < δ for all
s
s
t
s
t
s
t ∈ A.
12 Note
4
2. Representation Results on Subsets of Lotteries
The boundedness of the utility function in the von Neumann–Morgenstern representation in Theorem 1.2
might sometimes be restrictive. In order to deal with unbounded14 utility functions and monotonicity concepts such as the second stochastic order, we have to restrict the lotteries to subsets of M1 . Note however
that a von Neumann–Morgenstern representation might not exist on arbitrary subspaces as indicated by
the following counter-example which is a modification of Example 2.26 in F ÖLLMER AND S CHIED [8].
Example 2.1. As a subset of M1 (]0, +∞[), consider
D := µ ∈ M1 (]0, +∞[) lim t2 µ([t, +∞[) exists and is finite .
t→+∞
Define on D the function U (µ) := limt→+∞ t2 µ([t, +∞[) which is affine. Moreover, µ1 Q1 µ2 implies
that µ1 ([t, +∞[) ≥ µ2 ([t, +∞[) for all t ∈ ]0, +∞[ from which it follows that U is also monotone
with respect to the first stochastic order. Nevertheless, U does not admit a von Neumann–Morgenstern
representation since U (δx ) = 0 for any x ∈ ]0, +∞[. Furthermore, U (µ) = 0 for all µ ∈ D with
compact support, which in view of Lemma C.1 implies that U is not continuous with respect to the
variational norm k · k.
♦
Considering the special subset M1,c ⊂ M1 of lotteries with compact support15 we obtain a similar
representation result as Theorem 1.2 in terms of not necessarily bounded utility functions. Here, we
denote by B := B (S) the functions u : S → R of finite variation which may be unbounded.
Proposition 2.2. Let < be an affine preference order on M1,c monotone with respect to the first stochastic order. Then, the preference order < is σ (M1,c , B)-continuous and it admits a von Neumann–
Morgenstern representation
Z
U (µ) =
u dµ,
µ ∈ M1,c ,
(2.1)
for a nondecreasing utility function u : S → R.
Proof, Appendix C.3.
For lotteries with compact support we can also express the previous Proposition 2.2 under the assumption of monotonicity with respect to the second stochastic order Q2 defined for any µ, ν ∈ M1,c by
Z
Z
2
µ Q ν ⇐⇒
u dµ ≥ u dν for any nondecreasing concave u ∈ C,
where C := C (S) is the set of continuous functions u : S → R.
Corollary 2.3. Let < be an affine preference order on M1,c which is monotone with respect to the second
stochastic order. Then < is σ (M1,c , C)-continuous and admits a von Neumann–Morgenstern representation
Z
U (µ) = u dµ, µ ∈ M1,c ,
(2.2)
for a nondecreasing continuous concave utility function u : S → R. In particular, any µ ∈ M1,c has a
certainty equivalent, that is, there exists c ∈ S such that δc ∼ µ.
14 Indeed,
15 A
boundedness is for instance not ensured for concave utility functions on the real line.
lottery µ ∈ M1 has a compact support if there exists a compact K ⊂ S with µ (K) = 1.
5
Proof Appendix C.4.
Finally, we study special subsets of M1 for which the ψ’s moment exist. Following [8, Appendix A.6],
we fix a so called gauge function, that is, a continuous function ψ : S → [1, +∞[ and define




Z
ψ
ψ
dµ
<
+∞
.
(2.3)
(S)
=
µ
∈
M
:=
M
Mψ
1
1
1


The set Mψ
spans the vector space caψ of all signed measures µ ∈ ca for which the ψ’s moment exists,
R 1
that is, ψ |dµ| < +∞. By Bψ = Bψ (S) we denote the vector space of all functions u : S → R with
finite variation for which there exists a constant c such that |u(x)| ≤ c · ψ(x) for all x ∈ S. On this set we
consider the σ Mψ
1 , Bψ -topology, that is the coarsest topology for which for all f ∈ Bψ the mapping
Z
µ 7→
f dµ
is continuous. For instance, for ψ = max(1, |x|), the set Mψ
1 (R) consists of all lotteries on R with finite
first moment. Note that for ψ ≡ M for some constant M ∈ [1, +∞[, we have Mψ
1 = M1 and Bψ = Bb .
On caψ we also consider the ψ-variational norm given by
Z
kµkψ := kψ dµk =
ψ |dµ| ,
µ ∈ caψ ,
where ψ dµ ∈ ca denotes the signed measure with Radon-Nikodym derivative16 ψ with respect to µ.
Proposition 1.1 translates in the present context as follows.
Proposition 2.4. Let < be an affine preference order on Mψ
1 which is monotone with respect to the first
stochastic order. Then, the preference order < is k·kψ -continuous. In particular, any affine numerical
representation U of < is k·kψ -continuous, and there exists M > 0 such that
|U (µ) − U (ν)| ≤ M kµ − νkψ ,
(2.4)
for any µ, ν ∈ Mψ
1.
Proof, Appendix B.2.
On Mψ
1 we have the following representation result.
Theorem 2.5. Let < be an affine preference order on Mψ
1 monotone with respect to the first stochastic order. Then, the preference order < is σ(Mψ
,
B
)-continuous
and it admits a von Neumann–Morgenstern
ψ
1
representation
Z
U (µ) =
u dµ,
µ ∈ Mψ
1,
for some nondecreasing utility function u : S → R satisfying |u| ≤ c · ψ for some constant c.
Proof, Appendix C.5.
16 For
the Radon-Nykodym theorem, see [11, Theorem 2.10].
6
(2.5)
A. Notations
Throughout, S is a real interval. As mentioned before, M1 := M1 (S) and ca := ca (S) are respectively
the space of probability measures and signed measures on the Borel σ-algebra of S. The space ca is the
span of M1 . The first stochastic order Q1 on ca is defined as
Z
Z
µ Q1 ν ⇐⇒
u dµ ≥ u dν for all nondecreasing u ∈ Cb ,
where Cb is the set of bounded continuous functions u : S → R.
The variational norm k·k on ca is given by
)
(
X
kµk := sup
|µ (An )| A0 , A1 , . . . is a Borel partition of S ,
µ ∈ ca.
n
The weak topology σ (M1 , Y) on M1 is induced by the weak topology σ (ca, Y) on ca, that is, the
coarsest topology for which the linear functionals
Z
µ 7→
u dµ
are continuous for all u ∈ Y, where either Y = Bb , the set of bounded functions u : S → R with finite
variation, or Y = Cb ⊂ Bb .
B. Banach Lattice Structure for the First Stochastic Order
Given a convex set X , an affine function U : X → R satisfies for all x, y ∈ X and λ ∈ ]0, 1[
U (λx + (1 − λ) y) = λU (x) + (1 − λ) U (y) .
In the following V denotes the linear span of the set X − X . Note that X is not necessarily a subset of V
unless 0 ∈ X . A similar argumentation as in the proof of Lemma B.1 below shows that
n
o
V = λ (ν − η) ν, η ∈ X and λ ≥ 0 .
A preorder Q on X is a vector preorder if there exists a convex cone17 K ⊂ V with 0 ∈ K such that for
any µ, ν ∈ X ,
µ Q ν ⇐⇒ µ − ν ∈ K.
The same order is then also used on V. A function U : X → R is monotone with respect to the vector
preorder Q if U (µ) ≥ U (ν) whenever µ Q ν.
Lemma B.1. Given an affine function U : X → R monotone with respect to a vector preorder Q, there
exists a linear function Û : V → R also monotone with respect to Q such that for some µ0 ∈ X
Û (µ − µ0 ) = U (µ) − U (µ0 ) ,
17 A
µ ∈ X.
set K ⊂ V is a convex cone if K is convex and λx ∈ K for any x ∈ K and all λ > 0.
7
(B.1)
Proof. We define
Û (µ) := λ (U (ν) − U (η))
for µ := λ (ν − η) with ν, η ∈ X and λ ≥ 0. It is well-defined since for µ := λ (ν − η) = λ0 (ν 0 − η 0 )
with ν, η, ν 0 , η 0 ∈ X and λ, λ0 > 0
λ
λ0
λ
λ0
0
ν
+
η
=
η
+
ν0 ∈ X
λ + λ0
λ + λ0
λ + λ0
λ + λ0
which, by means of the affinity of U , yields λ (U (ν) − U (η)) = λ0 (U (ν 0 ) − U (η 0 )). Showing that
Û is linear and that (B.1) holds, follows analogously. It remains to show that Û is monotone. Since Û
is linear and Q is a vector preorder, it is enough to show that Û (µ) ≥ 0 for any µ Q 0. Given such a
µ = λ (ν − η) Q 0 for some ν, η ∈ X and λ ≥ 0, it follows that ν − η ∈ K, since K is a convex cone.
From the monotonicity of U follows U (ν) ≥ U (η) implying that Û (µ) ≥ 0.
Lemma B.2. If X = M1 , then V := span (X − X ) = {µ ∈ ca | µ (S) = 0}.
Proof. The case ⊂ is obvious. Conversely, for µ ∈ ca with µ (S) = 0, the Jordan-Hahn decomposition18
yields µ = µ+ − µ− for µ+ , µ− ∈ ca+ , where ca+ denotes the subspace of nonnegative measures in ca.
Without loss of generality, µ 6= 0, and therefore λ := µ+ (S) = µ− (S) > 0. Hence, µ = λ (ν − η)
where ν = µ+ /λ ∈ M1 and η = µ− /λ ∈ M1 .
Lemma B.3. If X = M1 , then the first stochastic order Q1 is a vector preorder for



Z

K= µ∈V
udµ ≥ 0 for any nondecreasing u ∈ Cb


and V = K − K, that is, for any µ ∈ V, there exist µ1 , µ2 Q1 0 such that µ = µ1 − µ2 .
Proof. That Q1 is a vector preorder for the convex cone



Z

K= µ∈V
udµ ≥ 0 for any nondecreasing u ∈ Cb


is immediate from the definition of the first stochastic order. We are left to show that any µ ∈ V can be
decomposed in µ = µ1 − µ2 where µ1 , µ2 Q1 0. Denote by F (t) := µ (]−∞, t] ∩ S) the cumulative
distribution function of µ. From µ (S) = 0 follows F (−∞) = F (+∞) = 0. Since t 7→ F (t) has
bounded variation and is right-continuous, the same holds for F1 := max (F, 0) and F2 := max (−F, 0).
We can therefore define the signed measures µ2 := −dF1 and µ1 := −dF2 , which by construction satisfy
µ1 − µ2 = dF1 − dF2 = dF = µ
and µ1 (S) = µ2 (S) = 0 as Fi (−∞) = Fi (+∞) = 0 for i = 1, 2, showing that µ1 , µ2 ∈ V. Moreover,
µi Q1 0 which follows by integration by parts,
Z
Z
Z
u dµi = − u dFi = Fi du ≥ 0, for any nondecreasing u ∈ Cb ,
for i = 1, 2 since du is a nonnegative measure.
18 See
for instance [11, Theorem 2.8].
8
Proposition B.4. If X = M1 , then (V, k·k) is a Banach lattice19 for the first stochastic order.
Proof. The fact that V is a lattice is immediate from the previous lemmata. It is also a Banach space,
since ca is complete for the variational norm and V = {µ ∈ ca |µ (S)
closed in ca.
= R0} is obviously
R
1
2
1
Finally,
the decomposition of µ in µ and µ as in B.3, holds µ = |dF2 | ≤ |dF | = kµk and
for
R
R
µ2 = |dF1 | ≤ |dF | = kµk. Hence, from µ Q1 ν Q1 0 follows kµk ≥ kνk.
Theorem B.5. Given X = M1 and some linear function Û : V → R monotone with respect to the first
stochastic order, then Û is continuous with respect to the variational norm k·k.
Proof. Since (V, k·k) is a Banach space, the result follows from well-known automatic continuity results.
For the sake of completeness, we present a proof in this case. Suppose by way of contradiction that
Û is not continuous. Then, there exists a sequence µk ∈ V such that kµk k = 1 and Û (µk ) ≥ 2k .
According to Lemma B.3, this sequence decomposes in µk = µ1k − µ2k , where µik Q1 0 for i = 1, 2.
From the Banach lattice structure of V for the first stochastic order, see Proposition B.4, it follows that
P
µ := k≥1 2−k µ1k ∈ V. Since Û (µ1k ) ≥ 0 for all k ∈ N, we end up with
Û (µ) ≥
n
X
n
X
2−k Û µ1k ≥
2−k Û (µk ) ≥ n,
k=1
for all n,
k=1
in contradiction to the assumption that Û (µ) ∈ R.
B.1. Proof of Proposition 1.1
Proof. Let U be an affine numerical representation of the affine preference order < on M1 . Due to
Lemma B.1, there exists a monotone linear function Û : V → R and µ0 ∈ M1 such that
Û (µ − µ0 ) = U (µ) − U (µ0 ) ,
µ ∈ M1 ,
where V = span (M1 − M1 ). Due to Theorem B.5, Û is a continuous functional for the variational
norm k·k. Since U (µ) = Û (µ − µ0 ) + U (µ0 ), it follows
|U (µ) − U (ν)| = Û (µ − µ0 ) − Û (ν − µ0 ) ≤ kÛ k∗ kµ − νk , for any µ, ν ∈ M1 ,
for the operator norm kÛ k∗ := sup{Û (µ) | µ ∈ ca , kµk ≤ 1} and the proof is complete.
B.2. Proof of Proposition 2.4
Proof. Using the fact that µ ∈ caψ is equivalent to ψdµ ∈ ca where ψdµ is the measure in ca with RadonNikodym derivative with respect to µ equal to ψ, a similar argumentation as in the proof of Lemma B.2
and B.3 shows that they also hold in the case where X = Mψ
1 and K is replaced by



Z

K := µ ∈ Mψ
udµ
≥
0
for
any
nondecreasing
u
∈
C
b
1


ψ
and ca by caψ . As for Proposition B.4, again, direct inspection shows that V ψ := span Mψ
1 − M1 is
complete for the norm k·kψ . Hence, Theorem B.5 also holds in the case where X = Mψ
1 with the norm
k·kψ . The proof of Proposition 2.4 is then analogous to the proof of Proposition 1.1.
19 For
the definition of a Banach lattice we refer to Chapter 9 in [1].
9
C. Technical Proofs
C.1. Proof of Theorem 1.2
Before addressing the proof of Theorem 1.2, we need the following lemma.
Lemma C.1. The space of lotteries with compact support M1,c is k·k-dense in M1 .
Proof. Consider an increasing sequence Kn of compacts in S such that ∪n Kn = S. Take µ ∈ M1 and
for Kn big enough such that µ (Kn ) > 0. Define the lottery µn ∈ M1,c by
µn (A) := µ (A ∩ Kn ) /µ (Kn ) ,
for any Borel set A ⊂ S.
We further define the measures µ̃n and µ̄n by µ̃n (A) := µ (A ∩ Knc ) and µ̄n (A) := µ (A ∩ Kn ). Then
1 kµ − µn k ≤ kµ̄n − µn k + kµ̃n k ≤ 1 −
kµk + µ (Knc ) −−−−−→ 0
n→+∞
µ (Kn ) due to the σ-additivity of µ.
Proof (of Theorem 1.2). For any x ∈ S, we define u(x) := U (δx ) which is a bounded function. Indeed,
according to Proposition 1.1, there exists M > 0 such that
|u(x) − u(y)| = |U (δx ) − U (δy )| ≤ M kδx − δy k ≤ 2M,
for all x, y ∈ S.
By monotonicity with respect to the first stochastic order, u is moreover a nondecreasing function, and
therefore it has only a countable number of discontinuities.
Step 1. In this step, we suppose that S = [a, b] for a < b and that u is right-continuous in a and leftcontinuous in b. We show that for any µ ∈ M1 ([a, b])
Z
U (µ) = u dµ.
(C.1)
We denote u (x+) := inf {u (y) | y > x}, and u (x−) := sup {u (y) | y < x}, and by uc the continuous
part of u on [a, b] defined as
X X uc (x) := u (x) −
u (y+) − u (y) −
u (y) − u (y−) .
a≤y<x
a<y≤x
From the boundedness of u, for any ε > 0, there exists δ > 0 such that
X
u (y+) − u (y−) ≤ ε.
(C.2)
y∈]a,b[
|u(y+)−u(y−)|<δ
Fix some ε > 0, and according to relation C.2, denote by s1 , . . . , sNε the finite number of jump points of
u of size greater than δ, that is, the points20 {x ∈ ]a, b[ | u (x+) − u (x−) ≥ δ}. We then define
uε (x) :=uc (x) + ∆ε (x)
:=uc (x) +
Nε h
X
i
h
i
u (sk +) − u (sk ) 1]sk ,b] (x) + u (sk ) − u (sk −) 1[sk ,b] (x) .
k=1
20 Note
that a jump point here is a positive difference between the left and right-continuous version of u.
10
By definition of uε we have ku − uε k∞ < ε on [a, b].
We now consider some subdivision σn = {a = x0 , x1 , . . . , xn = b} of [a, b] such that |σn | ≤ 1/n, where
|σn | denotes the mesh of the subdivision. We also suppose that this subdivision is fine enough that each
jump point of u of size greater than δ on [a, b] is also part of this subdivision. Let i1 , i2 , . . . , iNε denote
the ordered indices of the subdivision such that xik = sk .
PNε
PNε
βk δsk and (C.1)
βk = 1, then µ = k=1
Given µ ∈ M1 ([a, b]), denote by βk = µ ({sk }). If k=1
PNε
trivially holds. If k=1 βk < 1, consider µc given by
1−
µ=
Nε
X
!
βk
µc +
k=1
Nε
X
βk δsk ,
k=1
which is a probability measure such that µc ({sk }) = 0 and therefore
lim µc ([sk , sk + h[) = 0,
k = 1, . . . , Nε .
h&0
We now define
µ̂n =
n−1
X
αr δxr ,
and
µ̌n =
r=0
n−1
X
(C.3)
αr δxr+1 ,
r=0
Where αr = µc ([xr , xr+1 [) if r < n − 2 and αn−1 = µc ([xn−1 , b]). By definition, it is obvious that
µ̌n Q1 µc Q1 µ̂n and therefore
Z
Z
u dµ̌n ≥
Z
u dµc ≥
u dµ̂n .
(C.4)
From the monotonicity of U with respect to the first stochastic order also holds
Z
Z
u dµ̌n = U (µ̌n ) ≥ U (µc ) ≥ U (µ̂n ) =
u dµ̂n .
(C.5)
Subtracting (C.4) from (C.5) yields
Z
Z
Z
Z
Z
U (µc ) − u dµc ≤ u dµ̌n − u dµ̂n = u dµ̌n − u dµ̂n .
(C.6)
Together with the affinity of U and the definition of u, it follows that
! Z
Z
Nε
X
U (µ) − u dµ = 1 −
β
U
(µ
)
−
u
dµ
k c
c
k=0
Z
Z
Z
Z
≤ u dµ̌n − u dµ̂n = (u − uε ) d (µ̌n − µ̂n ) + uε d (µ̌n − µ̂n )
Z
≤ 2ε +
11
Z
uc d (µ̌n − µ̂n ) +
∆ε d (µ̌n − µ̂n ) . (C.7)
R
R
We finally show that that uc d (µ̌n − µ̂n ) + ∆ε d (µ̌n − µ̂n ) can be arbitrarily small as the mesh of
the subdivision converges to 0 for n going to ∞. As for the first term to estimate, since uc is uniformly
continuous on the compact [a, b], it follows that
Z
0≤
uc d (µ̌n − µ̂n ) =
n−1
X
αr uc (xr+1 ) − uc (xr ) ≤
sup
|uc (xr+1 ) − uc (xr )| −−−−−→ 0.
n→+∞
r=0,...,n−1
r=0
By definition, the other term to estimate yields
Z
0≤
∆ε d (µ̌n − µ̂n ) =
Nε
n−1
XX
αr u (sk +) − u (sk ) 1]sk ,b] (xr+1 ) − 1]sk ,b] (xr )
r=0 k=1
+
Nε
n−1
XX
αr u (sk ) − u (sk −) 1[sk ,b] (xr+1 ) − 1[sk ,b] (xr ) .
r=0 k=1
However, the terms 1]sk ,b] (xr+1 ) − 1]sk ,b] (xr ) and 1[sk ,b] (xr+1 ) − 1[sk ,b] (xr ) are equal to 0 for any
r = 0, . . . , n except for r = ik where xik = sk in which case 1]sk ,b] (xr+1 ) − 1]sk ,b] (xr ) = 1 and for
r = ik − 1 where xik +1 = sk in which case 1[sk ,b] (xr+1 ) − 1[sk ,b] (xr ) = 1 . Since Nε does not depend
on n, it follows
Z
0≤
∆εl d (µ̌n − µ̂n ) =
Nε X
u (sk +) − u (sk ) αik + u (sk ) − u (sk −) αik −1
k=1
≤ 2Nε kuk∞
h
sup
µc ([sk , xik +1 [) + µc ([xik −1 , sk [)
i
k=1,...,Nε
≤ 2Nε kuk∞
h
sup
µc
k=1,...,Nε
i
1
1
+ µc sk − , sk
.
sk , sk +
n
n
By means of relation (C.3), holds
sup
µc
k=1,...,Nε
1
sk , sk +
n
−−−−−→ 0,
n→+∞
and since sk − n1 , sk −−−−−→ ∅, the σ-additivity of µc yields also
n→+∞
sup
k=1,...,Nε
µc
sk −
1
, sk
n
−−−−−→ 0.
n→+∞
Hence, for any ε > 0 and sufficiently large n holds
Z
U (µ) − udµ ≤ 3ε
showing (C.1).
Step 2. In this step, suppose now that S is an open interval. Since u has countably many jump points,
we can find an increasing sequence [am , bm ] of compact intervalls such that S = ∪n [an , bn ] where u is
continuous in each an and bn . Consider now some µ ∈ M1 and let us show that
Z
U (µ) = u dµ.
12
To this aim, fix some ε > 0. Since U is k·k-continuous by Proposition 1.1, it follows by means of Lemma
C.1 that there exist some µε ∈ M1,c such that
|U (µ) − U (µε )| ≤ ε/2,
and
Z
Z
u dµ − u dµε ≤ ε/2.
Fix m0 ∈ N such that µε ∈ M1 ([am0 , bm0 ]). Due to relation (C.1) it follows
Z
Z
Z
Z
U (µ) − u dµ ≤ |U (µ) − U (µε )| + U (µε ) − u dµε + u dµε − u dµ ≤ ε.
Step 3. We are left to show that the von Neumann-Morgenstern representation also holds in the case
where S is not open, for instance of the form [a, b[ for a, b ∈ R. The cases [a, b], [a, +∞[, ]a, b] and
] − ∞, b[ work analogously. For any µ ∈ M1 , there exist λ ∈ [0, 1] and ν ∈ M1 (]a, b[) such that
µ = λδa + (1 − λ) ν. By affinity it follows from the previous representation result that
Z
U (µ) = λu (a) + (1 − λ)
]a,b[
Z
u dν =
u dµ,
[a,b[
and this ends the proof.
C.2. Proof of Corollary 1.6
Proof. Let U be an affine numerical representation of <. Due to Theorem 1.2, U has a von Neumann–
Morgenstern representation
Z
U (µ) = u dµ, µ ∈ M1 ,
for some bounded nondecreasing utility function u : S → R. Let us show that u is right-continuous.
Suppose by way of contradiction that u is not right-continuous in t ∈ S. Then, there exists m ∈ R such
that u (t) < m < inf s>t u (s). Pick now t0 > t and λ ∈ [0, 1] such that m = λu (t) + (1 − λ) u (t0 ).
Hence
n
o
s ∈ S u (s) ≥ m = {s ∈ S | δs < λδt + (1 − λ) δt0 } = ]t, +∞[ ∩ S
which is relatively open in S in contradiction to (1.4). Thus, u is right-continuous. Since any bounded
nondecreasing right-continuous function is a pointwise limit from above of bounded continuous functions, the Lebesgue monotone convergence theorem implies that U is the limit of a decreasing sequence
of σ (M1 , Cb )-continuous affine functions, hence is σ (M1 , Cb )-upper semicontinuous.
C.3. Proof of Proposition 2.2
Proof. In case that δx1 ∼ δx2 for all x1 , x2 ∈ S, the result is obvious. Otherwise, pick c2 > c1
with δc2 δc1 . Consider a countable increasing sequence of closed intervals [an , bn ] ⊂ S such that
S
c1 , c2 ∈ [a0 , b0 ] and n [an , bn ] = S, implying that M1,c = ∪n M1 ([an , bn ]). The restriction of <
13
to M1 ([an , bn ]) fulfils the conditions of Theorem 1.2, hence, this restriction admits a von Neumann–
Morgenstern representation of the form
Z
Un (µ) =
un dµ,
µ ∈ M1 ([an , bn ]) ,
[an ,bn ]
for some bounded nondecreasing utility functions un : [an , bn ] → R. Let us now show that we can
construct a von Neumann–Morgenstern representation of <. Since any affine numerical representation is
unique up to strict affine transformation, the function un is uniquely determined if we fix un (c1 ) = 0
and un (c2 ) = 1 for all n ∈ N. This implies in particular that un = um on [am , bm ] for any n > m.
We then obtain a nondecreasing, not necessarily bounded, utility function u : S → R. Moreover, for any
µ ∈ M1,c , we have
Z
Z
un dµ = Un (µ) ,
U (µ) = u dµ =
[an ,bn ]
for any n such that µ ∈ M1 ([an , bn ]), and this ends the proof.
C.4. Proof of Corollary 2.3
Proof. If a preference order < is monotone with respect to the second stochastic order, then it is monotone with respect to the first stochastic order. Indeed, suppose that µ Q1 ν, then by definition holds
µ Q2 ν since any continuous concave nondecreasing function is a fortiori continuous nondecreasing and
therefore, µ < ν. Let U be an affine numerical representation for < which by means of Proposition 2.2
has a von Neumann–Morgenstern representation of the form
Z
U (µ) =
u dµ,
µ ∈ M1,c ,
for some continuous increasing utility function u. This utility function is concave since for any x, y ∈ S
and λ ∈ [0, 1], the lottery δλx+(1−λ)y dominates in the second stochastic order the convex combination
λδx + (1 − λ) δy and therefore
u (λx + (1 − λ) y) = U δλx+(1−λ)y ≥ U (λδx + (1 − λ) δy ) = λu (x) + (1 − λ) u (y) ,
which ends the proof.
C.5. Proof of Theorem 2.5
Proof. Let U be an affine numerical representation of < on Mψ
1 . Define u (x) = U (δx ) for x ∈ S. Due
to Proposition 2.4, U is k·kψ -continuous, hence
|u (x)| ≤ M kδx kψ + kδy kψ + |u (y)| = M kψdδx k + kψdδy k + |u (y)| ≤ M̃ ψ (x) ,
x ∈ S,
for some y ∈ S and constants M, M̃ > 0 where the last inequality holds since ψ ≥ 1. Hence, u ∈ Bψ .
R
Let us show now that U (µ) = u dµ for any µ ∈ Mψ
1 . Adapting the proof of Lemma C.1, M1,c is
14
k·kψ -dense in Mψ
1 . Moreover, for any compact K ⊂ S, the norms k·k and k·kψ are obviously equivalent
on M1 (K) ⊂ Mψ
1 . We can therefore apply Proposition 2.2 to get
Z
U (µ) =
u dµ,
K
for any µ ∈ M1,c (K). A similar argumentation as in the second step in the proof of Theorem 1.2 ends
the proof.
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